Before examining long-range connectivity, we revisit the Hofstadter model

Fig 1  Multichannel signal transport performance in the Hofstadter model.

a A schematic of a coupled-resonator square lattice with nearest-neighbor interactions between pseudospin modes (σ = +1). b The Hofstadter butterfly, representing the gap Chern numbers C with different colors. An example of the wing segment for C = 3 is illustrated with the red outlines. The color bar represents C. c The tradeoff between C and A_C,i. The red and blue lines denote the C-A_C,i relationships for the largest and smallest wings, respectively. The gray dots depict the areas of intermediate-sized wings

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where ω₀ is the resonance frequency of each resonator, $$a_{n,\sigma }^{†}$$ and a_n,σ are the creation and annihilation operators of the pseudospin mode σ = ±1 at the nth site, respectively, t and φ_mn are the amplitude and phase of the interaction between the mth and nth sites, respectively, and 〈m, n〉 denotes the summation indices over all the nearest-neighbor resonator pairs. To realize broken time-reversal symmetry for a pseudospin mode σ = +1, nearest-neighbor interactions are designed to impose the synthetic flux Φ = +2πα on the photons during a counterclockwise rotation around each unit cell. The flux for the square lattice (Fig. 1a) is achieved with the following hopping phases using the Peierls substitution

where (m₁, m₂) and (n₁, n₂) are the integer indices denoting the positions of the mth and nth sites, respectively, and δ_a,b is the Kronecker delta function. The designed hopping phase can be implemented with non-resonant waveguide loops

The topological invariant determining the number of topologically protected edge modes

where C_l(p,q) is well-defined except for gap closing with even q and l = q/2. The colored Hofstadter butterfly

According to Eq. (3), C_l can have an arbitrary integer by controlling the flux parameters p and q, and the bandgap number l, allowing any numbers of topologically protected edge modes within the frequency range of ω₀ – 4t < ω < ω₀ + 4t. However, increasing the number of edge modes substantially decreases the bandgap width (Δω in Fig. 1b) and the allowed range of synthetic fluxes (Δα in Fig. 1b). This tradeoff between C_l and the parameter ranges Δω and Δα limits the signal transport performance in multichannel applications. For example, the operation bandwidth of the edge modes is restricted by Δω, and the allowed range of Δα determines the robustness of the flux, which is very sensitive to optical phases.

To quantify the tradeoff relation, first we investigate the number of butterfly wings—or topological bandgaps—N_C for a specific gap Chern number C ≡ C_l: N_C = φ(1) + ··· + φ(2 | C |) for the Euler's phi function φ(n) (Supplementary Note S3). The performance of C-channel signal transport using topologically protected edge modes can then be characterized by the area A_C,i of the ith wing (i = 1, 2, …, N_C) for each C, where

Δω(α) is the bandgap width at α, and α_C,i^U and α_C,i^L are the upper and lower bounds of the wing, respectively, satisfying the gap closing at Δω(α_C,i^U) = Δω(α_C,i^L) = 0. We numerically calculate A_C,i for C ≤ 15 by approximating each wing as a 60-gon (Supplementary Note S3).

Figure 1c shows the tradeoff relation between the channel number C and the signal transport performance A_C,i. Notably, we reveal the i-dependent transition between the power law and exponential distributions between C and A_C,i. First, the largest wings (i = 1) satisfy the power law A_C,1 ∝ C^–1.95 (red solid line in Fig. 1c). This quadratic scaling originates from C-dependencies of Δα = α_C,i^U – α_C,i^L and Δω: Δα is equal to C^–1/2, and Δω is approximately proportional to C^–1 due to the convergence to the Landau levels

To overcome the limit of the signal transport performance A_C,i, we exploit long-range connectivity preserving discrete translational symmetry. We focus on a class of the long-range connectivity that can be decomposed into the decoupled overlap of the translated lattices having only nearest-neighbor interactions. For example, overlapping the identical lattices with in-plane translation allows for the coexistence of their eigenstates. Because Δω and Δα maintain their values of an individual lattice due to decoupling, the increase of the gap Chern number C is expected while preserving the performance figure A_C,i. Importantly, the vertical stacking of identical two-dimensional (2D) systems, which has recently become a substantial idea for achieving high Chern numbers in magnetic topological insulator films

To employ the overlap of lattices, we apply the following two criteria for real implementation. First, the sites and connections of the lattices should not be overlapped to maintain the band properties of the original lattices. Second, the pointwise crossings between the connections are allowed by employing the waveguide crossing structures with negligible crosstalks and losses. We demonstrate our proposal by overlapping two lattices as an example. Figure 2a shows two decoupled square lattices possessing nearest-neighbor interactions with the same lattice constant d. In each lattice, we impose the hopping phase of the Hofstadter model with the same α. The lattices are 45° rotated from the original Hofstadter lattice (Fig. 1a) while maintaining connectivity. When the lattices are overlapped in the 2D plane by translating a lattice d/2^1/2 along the x-axis, the obtained overlapped square lattice has next-nearest-neighbor interactions with nearest-neighbor decoupling, realizing long-range connectivity.

Fig 2  Overlapped Hofstadter lattices for long-range connectivity.

a Schematics of the two decoupled square lattices having nearest-neighbor interactions (left) and the overlapped lattice having only next-nearest-neighbor interactions (right). Orange circles and gray lines denote the site resonators and non-resonant waveguide couplers. The lattice constant of each composing lattice is d. The black box on the overlapped lattice indicates the crossing between long-range interactions. b The silicon waveguide implementation

Reference 38

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38 of the black box region in (a). The inset shows the FDTD simulation result for the waveguide crossing region. D_r, D_c, d_g, d_c, r_f, l_c and t_c represent geometric parameters (Materials and methods for their values). c The maximum wing area A_C,1 with respect to the gap Chern number C for different numbers of the overlaps N. The Hofstadter butterflies for N = 2 (d) and N = 3 (e). Color bars represent C

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The Hamiltonian of the overlapped lattice is given by the direct sum as:

where H₁ and H₂ are the Hamiltonians of the Hofstadter model represented by Eq. (1), corresponding to the top and bottom square lattices in Fig. 2a, respectively. Because H₁ and H₂ possess identical band structures, the Hamiltonian H leads to the degeneracy of the original bands. The generalization of the two-lattice overlap toward the overlap of an arbitrary number of lattices can also be described with the direct sum: H = ⨁_nH_n.

The hurdle in realizing the overlapped lattice is the crossings between long-range interactions (black box in Fig. 2a). To resolve this issue, we employ a waveguide crossing design to a non-resonant waveguide loop (Fig. 2b). By applying conventional system parameters in silicon photonics

To demonstrate our proposal, we calculate the maximum wing area A_C,1 for different numbers of the overlapped lattices (Fig. 2c). The increase of A_C,1 is directly proportional to the number of overlaps N, while preserving the power law behavior with the same exponent. Figure 2d, e illustrate the Hofstadter butterflies of two- and three-folded lattices, respectively, proving that the gap Chern numbers of an N-folded lattice domain are N times the original gap Chern number due to the degeneracy of the decoupled lattice overlap. We also note that the overlap of disparate lattices allows for any gap Chern number other than the multiples of N (Supplementary Note S5).

To explore nontrivial topological natures of overlapped lattices, we investigate the edge modes at lattice boundaries. We compare three different configurations: the surface of a bulk (Fig. 3a) and the interfaces between the bulks having different (Fig. 3b) and same C's (Fig. 3c). To obtain dispersion relations, we employ the one-dimensional (1D) ribbon geometry, which is infinite along the x-axis with the finite y-axis thickness.

Fig 3  Edge mode analysis.

a Field profile of the topologically nontrivial edge mode in the overlapped lattice. The domain has α = 1/3 for each sublattice, leading to C = 2 in the lower gap. The black arrow indicates the propagating direction of the mode. b Field profile of the topologically nontrivial edge mode at the interface between two overlapped lattices. The adjacent domains have inhomogeneous Chern numbers (C_bottom = 2 and C_top = −1). a, b denote the star markers in (d, e), respectively. c The absence of the topologically nontrivial edge mode at the interface between two overlapped lattices with homogeneous Chern numbers (C_bottom = C_top = 1). d−f The band structures of the systems in (a−c), respectively. k_x denotes the x-axis component of the reciprocal vector in the magnetic Brillouin zone. y_mean denotes the y-axis component of the center-of-mass of the corresponding modes. The dashed lines in (d) represent the ensemble averaged dispersions under 500 realizations of uniformly random diagonal disorder [–U_diag, +U_diag] on the resonance frequency, where U_diag = 0.5t. All the results are obtained with the tight-binding equation with the supercell technique (Supplementary Note S6)

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Figure 3a shows a finite bulk domain of a doubly overlapped lattice. A sublattice composing the domain has the synthetic flux of α = 1/3, leading to two bandgap openings with C = ± 1. The corresponding band structure is shown in Fig. 3d (Supplementary Note S6 for its calculation). The overlapped lattice composed of two identical sublattices provides doubly degenerate edge modes, which are apparent when diagonal disorder is applied (red dashed lines in Fig. 3d).

Figure 3b, e, c, f show the C-dependency of edge modes by examining the overlapped lattices having inhomogeneous and homogeneous C profiles, respectively. In Fig. 3b, we impose different synthetic fluxes on the bottom (α = 1/3) and top (α = −1/3) lattices. Due to the symmetric profiles of the Hofstadter butterflies for the overlapped lattices (Fig. 2d, e), the bandgaps of the bottom and top lattices fall within the same frequency range for the same |α|. Therefore, the given configuration in Fig. 3b facilitates the full use of the bandgap for signal bandwidths. Meanwhile, the gap Chern numbers of the bottom and top lattices at the first bandgap are disparate: C_bottom = 2 and C_top = −1, respectively. Therefore, three topologically protected edge modes are obtained from |C_bottom – C_top | = 3 (Fig. 3e). In contrast, C-homogeneous bulks in Fig. 3c, which are achieved with α = 1/3 at both bottom and top overlapped lattices, lead to the absence of topologically nontrivial edge modes (Fig. 3f).

Edge mode dispersions in Fig. 3e, f can be illustrated through the co- and counter-directional coupling between the edge modes in the isolated bulks, where the number of the modes is directly determined by C. For example, the interface in Fig. 3b can be understood as the junction of three planes: a plane with C = –1 and two planes with C = +1, resulting in the coupling of three co-propagating edge modes (solid arrows in Fig. 3b). On the contrary, the structure with homogeneous C in Fig. 3c results in the counter-propagating edge modes (dashed arrows in Fig. 3c), leading to the stark anti-crossing and the following transition from topologically nontrivial to trivial interface states (Fig. 3f).

To confirm the enhanced information capacity in signal transport through overlapped topological lattices, we analyze the scattering of edge modes. We examine the system that includes three interfaces between the overlapped lattices, thereby composing an edge-mode beam splitter (Fig. 4a, b). All the sublattices for the lattice overlap are designed with α = ±1/3, where the overlap derives different numbers and chirality of edge modes. To examine the scattering at the junction of the topological beam splitter, we calculate the frequency-dependent scattering matrix S_ij(ω) between the ith and jth edge modes within the bandgap, using the open-source Python package Kwant

Fig 4  Incoherent multichannel beam splitting.

a A schematic of the scattering region. Three domains possess α = ±1/3 to support the gap Chern number ±1 at the shared bandgap range. A 101 × 101 scattering region is assumed for the analysis in (c−e). b The scattering region with semi-infinite ports for applying the scattering matrix method. The orange arrows represent the centered edge modes of each port. The black arrows denote the outer edge modes resulting from finite widths of the ports. The incoherent transmittance is proportional to the number of centered edge modes. c Intensity profile for an incoherent light at the frequency ω = ω₀ − 1.5t. The input is obtained with 100 random linear combinations of the input edge modes. d Intensity profile against diagonal disorder. The resonance frequency of each site in the scattering region is perturbed uniformly between [−U_diag, +U_diag]. e The transmittances for the incoherent incidence against diagonal disorder. The transmittance curves for 50 realizations of diagonal disorder are overlaid. U_diag = 1.5t in (d, e). The shaded region in (e) denotes the frequency range of perfect beam splitting operation against disorder

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To fully exploit the superior performance of the overlapped lattices—large A_C,i with broader bandgaps and multiple edge modes—we focus on topologically protected wave functionalities applied to spatio-temporally incoherent light sources. The light source is defined by the superposition of the edge modes having uniformly random amplitudes and phases. To quantify the energy flow throughout the kth port (k = 1, 2, 3), we define incoherent transmittance T_k as the average of the squares of the scattering coefficients between port 1 and port k:

where N_in,1 is the number of the input modes at port 1, N_out,k is the number of the output modes throughout port k. Because the centered edge modes (orange arrows in Fig. 4b) barely overlap with the outer modes (black arrows in Fig. 4b), the sub-scattering matrix consisting of the centered edge modes is almost unitary.

From the unitary sub-scattering matrix, T_k in the frequency region supporting the centered edge modes is solely determined by the number of centered edge modes under the incoherent incidences (Fig. 4c), regardless of the phase matching condition at the splitter junction. Due to topologically nontrivial natures, the beam splitting is protected under different forms of disorder: diagonal (Fig. 4d) and hopping phase disorders (Supplementary Note S8). As shown in Fig. 4e, the beam splitting is maintained over the bandgap range with backscattering suppression, demonstrating incoherent functionality for multichannel integrated photonics for the first time.

We implement the overlapped lattice composed of ring resonators, which are pair-wisely coupled via off-resonant waveguide loops. A pair of coupled ring resonators is described by the following tight-binding Hamiltonian

where a^†, a, b^†, and b are the creation and annihilation operators of the resonator 'a' and 'b', ω₀ is the resonance frequency, τ is the energy leakage rate from each resonator to the evanescently coupled waveguide, and φ is the tunable phase shift obtained through the waveguide loop (Supplementary Note S1 for derivation from the temporal coupled mode theory).

We color the Hofstadter butterflies in Figs. 1c, 2d, e on a pixel-by-pixel basis with a 1440 × 1920 resolution. For each α = i/1920 (0 ≤ i < 1920), we numerically calculate the band structure of the Hofstadter Hamiltonian of Eqs. (1) and (2), discretizing the repeated part of the magnetic Brillouin zone as 20 × 20 points (Supplementary Note S2 for details). We determine the Chern number using the TKNN formula, Eq. (3), for each of the 1440 frequencies within the range (ω₀ − 4t, ω₀ + 4t).

We perform the FDTD full-wave analysis using Tidy3D

To extract the scattering matrix between the edge modes at the center junction in Fig. 4a, we extend the ports of the system to support the edge modes propagating from and to infinitely away while satisfying impedance matching. Three semi-infinite ports (Fig. 4b) are constructed by repeating the supercells of the overlapped lattices (Supplementary Note S6). We utilize the open-source python package Kwant